Unique continuation and the regularity of elliptic PDEs and generalized minimal submanifolds

椭圆偏微分方程和广义最小子流形的唯一延拓和正则性

• 批准号: 2350351
• 负责人: Zihui Zhao
• 金额: $ 25.37万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350351&HistoricalAwards=false
• 关键词: Unique; continuation; regularity; elliptic; PDEs

This award supports research on the regularity of solutions to elliptic partial differential equations and regularity of generalized minimal submanifolds. Elliptic differential equations govern the equilibrium configurations of various physical phenomena, for instance, those arising from minimization problems for natural energy functionals. Examples include the shape of free-hanging bridges, the shape of soap bubbles, and the sound of drums. Elliptic differential equations are also used to quantify the degree to which physical objects are bent or distorted, with far-reaching implications and applications in geometry and topology. The proposed research focuses on the regularity of solutions to such equations. Questions to be addressed include the following: Do non-smooth points (singularities) exist? How large can the set of singularities be? What is the behavior of the solution near a singularity? Is it possible to perturb the underlying environment in order to eliminate the singularity? The project will also provide opportunities for the professional development of graduate students, both via individual mentoring and via the organization of a directed learning seminar on geometric analysis and geometric measure theory.The mathematical objectives of the project are twofold. First, the principal investigator will study unique continuation for solutions to elliptic partial differential equations, with a focus on quantitative estimates on the size and structure of the singular set of these solutions. A second topic for consideration is the regularity theory for generalized minimal submanifolds (a generalized notion of smooth submanifolds which arise as critical points for the area functional under local deformations). In particular, the principal investigator will study branch singular points in the interior as well as at the boundary of a generalized minimal submanifold, under an area-minimizing or stability assumption. Research on the latter topic, which can be viewed as a non-linear analogue of quantitative unique continuation for elliptic equations, requires the integration of ideas from geometric measure theory, partial differential equations and geometric analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项支持有关椭圆形部分微分方程解决方案的规律性的研究以及广义最小次曼菲尔德的规律性。椭圆形微分方程控制了各种物理现象的平衡构型，例如，自然能量功能最小化问题的平衡构型。例子包括自由悬挂桥的形状，肥皂气泡的形状和鼓声。椭圆形的微分方程也用于量化物理对象弯曲或扭曲的程度，具有深远的含义以及几何和拓扑的应用。拟议的研究集中于此类方程的规律性。要解决的问题包括以下内容：是否存在非平滑点（奇异点）？一组奇点可以有多大？奇异性附近的解决方案的行为是什么？是否有可能扰动基础环境以消除奇异性？该项目还将通过个人指导和组织有关几何分析和几何测量理论的有指导学习研讨会的组织为研究生的专业发展提供机会。项目的数学目标是双重的。首先，首席研究者将研究椭圆形偏微分方程解决方案的独特延续，重点是对这些解决方案奇异集的大小和结构的定量估计。要考虑的第二个主题是广义最小亚曼叶的规律性理论（平滑的亚策略的普遍概念，在局部变形下作为该区域功能的关键点出现）。特别是，在面积最小化或稳定性假设下，主要研究者将研究内部和广义最小亚曼福尔德的边界的分支单数点。对后一个主题的研究可以看作是椭圆方程的定量独特延续的非线性类似物，就需要从几何措施理论，部分差分方程和几何分析中整合思想。该奖项反映了NSF的法规使命，并认为通过基金会的知识优点和广泛的crietia criter criter criter criter criter criter criter criter criter criter criter criter critia criter critia criter critia criter criter critia criter critia critia critia critia critia criteria crietia crietia均值得一提。